Which one of the following is correct? Ifz and w are complex numbers and `overset(-)(w)` denotes the conjugate of w, then `|z+w|=|z-w|` holds only, if

Options

A. `z=0or w=0`
B. `z=0and w=0`
C. `z.vecw` is purely real
D. w is purely imaginary

Explanation

To solve the problem, we need to analyze the condition under which the equation \( |z + w| = |z - w| \) holds true for complex numbers \( z \) and \( w \).

### Step-by-Step Solution:

1. **Define the Complex Numbers**:
Let \( z = x + iy \) and \( w = a + ib \), where \( x, y, a, b \) are real numbers.

2. **Express the Moduli**:
We need to evaluate \( |z + w| \) and \( |z - w| \):
\[ |z + w| = |(x + a) + i(y + b)| = \sqrt{(x + a)^2 + (y + b)^2} \]
\[ |z - w| = |(x - a) + i(y - b)| = \sqrt{(x - a)^2 + (y - b)^2} \]

3. **Set the Moduli Equal**:
We set the two expressions equal to each other:
\[ \sqrt{(x + a)^2 + (y + b)^2} = \sqrt{(x - a)^2 + (y - b)^2} \]

4. **Square Both Sides**:
Squaring both sides to eliminate the square roots gives:
\[ (x + a)^2 + (y + b)^2 = (x - a)^2 + (y - b)^2 \]

5. **Expand Both Sides**:
Expanding both sides results in:
\[ (x^2 + 2ax + a^2 + y^2 + 2by + b^2) = (x^2 - 2ax + a^2 + y^2 - 2by + b^2) \]

6. **Simplify the Equation**:
Canceling out common terms \( x^2, a^2, y^2, b^2 \) from both sides, we have:
\[ 2ax + 2by = -2ax - 2by \]

7. **Rearranging the Terms**:
Rearranging gives:
\[ 4ax + 4by = 0 \]
or simplifying further:
\[ ax + by = 0 \]

8. **Conclusion**:
The condition \( |z + w| = |z - w| \) holds true if \( ax + by = 0 \). This means that the vectors represented by \( z \) and \( w \) are orthogonal in the complex plane.